B Confused about symmetry

The part I'm confused about - in order to have two distinct observations of a system you want to check for symmetry isn't something asymmetrical is required to distinguish them... as distinct observations.

If we say a system is completely closed and are restricted to refer only to that system how do we know we have observed the system across any interval that could prove symmetry or lack thereof - if the system actually has perfect symmetry? Wouldn't that just mean - there is no observable change. How would we mark the observation as started or complete?"

If we enlarge the observed system to include an asymmetric measurement against which we see our (original) system is symmetric (doesn't change) - logically haven't we actually only observed asymmetry in the enlarged system. Doesn't it seem odd to turn around and say "we observed symmetry!" in our original system. It's not that saying it doesn't make sense but it just seems odd. To me it seems more reasonable to say - jeez we can't tell anything at all without reference to some asymmetry.

This is part of what is driving me crazy trying to picture invariant everywhere physical laws based on observed or expected symmetries.

I don't understand the question (it seems more general than Special or General Relativity)...
Why don't you [as an example] take the law of motion:
[itex] \frac{d^2 \vec{x}}{dt^2} = f(\vec{x}, \frac{d\vec{x}}{dt}.t)[/itex]
and try to tell me when that thing is invariant under rotations? What must not be true in order the above formula not to be invariant under a rotation?

I wasn't sure what forum to even put it under. I picked this one because of how symmetry is important to invariance.
And yes I do understand the practical importance, veracity, accuracy and beauty of the use of symmetry in describing mechanics.

I can read it: The rate of change of velocity vector of a thing is a function of its position vector, the sensitivity of its position vector to time, and time.
But the puzzle of Norton's dome is one that I think captures my question w/respect to that classical laws of motion. Take a sphere and put it at the very top of a hemispherical dome. It's position vector as a function of time is... what?

But to take it even further, say we are talking about a sphere. We want observe it's properties so we need an experiment. Our experiment needs to be defined as started and stopped, begun and ended, complete and subject to review. Let's say our lab is very sophisticated and our lab clock is perfectly mechanically connect to a button on the sphere (it is everywhere on the sphere) that says "rotate me".

So our clock pushes the button. Nothing happens that we can see. So we conclude the sphere rotated but there are pervasive, symmetrical (timeless) physical laws giving it symmetry under rotation... just seems to me an odd assertion. Even if we trust the button does what it says it does and all kinds of important symmetrical things happened when we clicked it, isn't it more precise to say that clocks (which are asymmetric counters if anything) are required to define and interval of change - when nothing observable is happening!

Then when rotating it back the idea that we have "put it back exactly the way it was" (it is unchanged by reversal) is even more strange. Now it seems to me we have confused proof of pervasive mechanisms of symmetry maintenance for a lack of observable change over two necessary changes in the clock. In other words it's not exactly the way it was, but we have no idea if it ever rotated at all - all we know is that our clock ("rotate" button clicker) advanced! As far as we know "rotate" means precisely nothing at all.

If you think of the clock as a compass-like mechanical thing (something must exist against which our experiment is identified) it seems even more clear - nothing can be said to "retain" symmetry because no experiment can be carried out that does not depend on reference to asymmetry - especially experiments where the observed system doesn't seem to change.

What I am struggling with here is philosophical, so I apologize for that, but I am laying on the floor over and over because I've tripped over core assumptions that seem to underlie the ontological debate at the core of QM and GR (lattices and QED etc). I can't see where the idea of everywhere continuous, reversible, symmetrical fields etc. come from - and why that metaphor dominates ontology over and above the idea of irreversible process (like Cantor middle third erasure for example) - which seems pretty much the opposite. It's not like either can escape the problem of infinite regress but the ether-like fields seems more invented than infinite irreversible process - which seem pretty much... life-like.

And I'm not just making this up to bother people on this forum. Swear to god. And I'm not a sock puppet for some fringe website. I am currently reading Smolin and Unger's book, "The Singular Universe and The Reality of Time". I would challenge anyone who has read that to explain to me where in the universe timeless symmetries are supposed to be actively governed. I think (maybe incorrectly) their point is that fundamentally there is only time and it is by definition asymmetrical - hence there is only process.

If we say a system is completely closed and are restricted to refer only to that system how do we know we have observed the system across any interval that could prove symmetry or lack thereof - if the system actually has perfect symmetry? Wouldn't that just mean - there is no observable change. How would we mark the observation as started or complete?"

Do we need to and if we do, what should we do when the symmetry was perfect but the system never completed one complete observable cycle? In calculus if you have an integral that does not converge at its limits you have an indefinite integral and regard the result as undefined while if the integral does converge at its limits you can call it an improper integral and can use the result. If one cycle is never completed can the integral still be considered improper and symmetric?

Nina Byers wrote a very informative paper in 1998 titled "E. Noether’s Discovery of the Deep Connection Between Symmetries and Conservation Laws:". Unfortunately the normal arXiv link in the PDF of the paper links to a different paper called "Dimensional Reduction" for some reason so try http://xxx.lanl.gov/abs/physics/9807044.

In the early days, Hilbert wrote about this problem as ‘the failure of the energy theorem ’. In a correspondence with Klein [3], he asserted that this ‘failure’ is a characteristic feature of the general theory, and that instead of ‘proper energy theorems’ one had ‘improper energy theorems’ in such a theory. This conjecture was clarified, quantified and proved correct by Emmy Noether.

Staff: Mentor

in order to have two distinct observations of a system you want to check for symmetry isn't something asymmetrical is required to distinguish them... as distinct observations.

Technically, yes. If we take a sphere and rotate it, we have to have some marking on the sphere that moves in order for us to know that we rotate it. So the sphere technically isn't perfectly symmetrical--the mark is at some specific place.

However, when we talk about symmetry of physical laws, we mean something different. If the laws of physics are symmetric under rotation, that means we can rotate our lab any way we like--and we can use markings to show that we've rotated it--and we can run experiments and get the same results we got before we rotated the lab. So the fact that we can tell, using markings, that we did rotate the lab is what allows us to test the fact that the laws of physics don't change when we rotate the lab. The symmetry is manifested in the unchanging results of the experiments.

Similarly, if the laws of physics are unchanged under time translation, that means we can run experiments when the clock on the lab wall reads different times, and still get the same results. We can tell the experiments were run at different times because of the different clock readings; and that is what allows us to test the fact that the laws of physics are unchanged when we change the time at which we do the experiment.

As @PeterDonis explains, my thought experiment confused (yet again) the symmetry of the physical laws (scalar fields) giving the sphere a Lagrangian action, and the symmetry of the object. Thanks @PeterDonis for the usual clarity.

So this probably doesn't really belong in this forum... Maybe classical mechanics?

It's these scalar fields that are confusing me... these fabric-like entities that give the sphere the same configuration and dynamics at different times and under the symmetry group transformations of the lab (i.e the results seem the same if we do the experiment later, or we do the experiment over in a lab next door that is at a different angle). Good learning moment - I forget how repeatable the world is... even though it keeps freaking changing!

So I started reading the link to that interesting paper about the history of Noether's theorem - and now I'm stuck trying to understand Hamilton's Principle and D'Alambert-Lagrange Principle - and trying to relate it to Norton's Dome...

If the fields (the bearers of physical laws of space-time) are infinitesimally perfectly symmetric over time - shouldn't the sphere just sit there - all the virtual forces cancelling. Why instead does it start rolling down the dome in some direction at some point.

If "random quantum fluctuations" cause the sphere to begin to roll are those fluctuations an asymmetric character of the field. If they are symmetric (the fluctuations) wouldn't the sphere still sit there - maybe vibrate a bit?

If the sphere was a ping-pong and the dome really really big comparatively (so the top of it seems almost flat) would we see the ping pong ball sort of roll around in random way trying to make up it's mind about which way to fall? How much could we stretch out that poise?

If "random quantum fluctuations" cause the sphere to begin to roll are those fluctuations an asymmetric character of the field. If they are symmetric (the fluctuations) wouldn't the sphere still sit there - maybe vibrate a bit?

No. A fluctuation means the sphere's position isn't exactly at the equilibrium point; and once its position isn't exactly at the equilibrium point, it will roll downwards, because the equilibrium at that point is unstable. This is an example of what is called "spontaneous symmetry breaking": fluctuations plus the natural dynamics of the system can take a state that has the underlying symmetry (in this case, rotational symmetry) to a state that doesn't.

The key thing to bear in mind about this is that, if there is an underlying symmetry to the system, then any individual state that does not have that symmetry will be one of a set of states that, taken as a whole, does. In other words, the symmetry still applies to the set of all the possible states of the system, even if it doesn't apply to one particular state. For example, in the case of the sphere rolling down the hill, if there is rotational symmetry present, the shape of the hill itself will be rotationally symmetric, so the set of states at the bottom of the hill will be rotationally symmetric, taken as a whole. But the sphere will end up in only one of those states, so its state, taken by itself, will not be rotationally symmetric.

my question or confusion, even if I'm not using the right description of the thing that is thought to "cause symmetry" the conservation law or field is... how do we know there is a thing that has symmetry?

Why don't we say "things that seem exceedingly similar to the point of being almost indistinguishable seem to happen (and are causable) at different times and places"? I mean don't they have to actually be different things, events, because we are observing their similarity (symmetry) relative to some asymmetry...different experiment times, places, locations, angles. Or are we saying that "makes no difference" means "are the same thing or event". Why don't we require all observations to be labeled as unique no matter how similar they seem, as they surely must be to be different observations - And if they are all unique none prove symmetry laws exist. They only prove that similarity laws exist.

Staff: Mentor

Why don't we say "things that seem exceedingly similar to the point of being almost indistinguishable seem to happen (and are causable) at different times and places"? I mean don't they have to actually be different things, events, because we are observing their similarity (symmetry) relative to some asymmetry...different experiment times, places, locations, angles. Or are we saying that "makes no difference" means "are the same thing or event". Why don't we require all observations to be labeled as unique no matter how similar they seem, as they surely must be to be different observations - And if they are all unique none prove symmetry laws exist. They only prove that similarity laws exist.

The part I'm confused about - in order to have two distinct observations of a system you want to check for symmetry isn't something asymmetrical is required to distinguish them... as distinct observations.

If we say a system is completely closed and are restricted to refer only to that system how do we know we have observed the system across any interval that could prove symmetry or lack thereof - if the system actually has perfect symmetry? Wouldn't that just mean - there is no observable change. How would we mark the observation as started or complete?"

If we enlarge the observed system to include an asymmetric measurement against which we see our (original) system is symmetric (doesn't change) - logically haven't we actually only observed asymmetry in the enlarged system. Doesn't it seem odd to turn around and say "we observed symmetry!" in our original system. It's not that saying it doesn't make sense but it just seems odd. To me it seems more reasonable to say - jeez we can't tell anything at all without reference to some asymmetry.

This is part of what is driving me crazy trying to picture invariant everywhere physical laws based on observed or expected symmetries.

Here Murray Gell Mann explains about beauty truth and physics, and symmetry plays a fundamental role in his video. He is the one who discovered quarks.

Since the universe seems to be simpler when we understand it more fundamentally, generalizing symmetry is more reliable when the symmetry is more fundamental. It is because then it has the property which Gell Mann refers to as "beauty" which means simplicity. So based on this he published his theory of quarks despite that it disagreed with experiments, because it had the property of beauty, or simplicity. So the moral of the story is, symmetry is a useful way to derive fundamental physics, from our otherwise more macroscopic observations. This is how we get closer to the fundamental law as Gell Mann talks about.

Also to answer one of your other questions: Obviously the property of difference must be the case in the observation of symmetry, since if not you could only observe the very same thing. You observe a similarity, and then hypothetically generalize further than the difference in the symmetry to make a hypothesis.

He's pretty funny. Nice talk. I haven't read his book though I think I would enjoy it.

Last night/yesterday I realized that Nowak's "Evolutionary Dynamics" is what sent my head spinning a year and a half ago.

My question, the one I got from reading his book but can only ask now: Is spontaneous symmetry breaking the same thing (deeply symmetrical with) population fixing by neutral drift?

I think this is what bugs me about the language around fields. There is no "jaguar field" or "hyena field" why should we use "fields" to describe emergent entities at the sub-atomic level when we don't use them to describe any other emergent entities?

I may be stating the obvious and asking a dumb question, but maybe I'm just getting it.

Is the idea of symmetry groups not that some group of perfectly identical things exist because their fields are fundamental but that mapping groups is a way to see the fitness landscape and payoff matrix from which they emerge, just like classifying hyenas and jaguars? Clearly no two jaguars are the same jaguar but there is classification symmetry and it says something about the Savanna.

Staff: Mentor

Is spontaneous symmetry breaking the same thing (deeply symmetrical with) population fixing by neutral drift?

I can see the implicit analogy you're making here, but I'm not sure it's a very fruitful one. And there is one obvious disanalogy: in neutral drift, the "potential" (fitness) of different alleles of a gene is the same, whereas in spontaneous symmetry breaking, the potential of the final state is lower than that of the initial state.

There is no "jaguar field" or "hyena field" why should we use "fields" to describe emergent entities at the sub-atomic level when we don't use them to describe any other emergent entities?

One could just as well ask why we don't use the word "field" to describe jaguars or hyenas, since we use it in subatomic physics. Different areas of science use different language because they're talking about different things. The fact that there might be (strained) analogies between certain aspects of those things does not mean we should use the same word to describe them all. If you want a more general word to describe the category of things of which jaguars, hyenas, and subatomic quantum fields are all special cases, well, you just gave one: "emergent entities".

Clearly no two jaguars are the same jaguar but there is classification symmetry

Again, I think this is not a very fruitful analogy. Another obvious disanalogy is that there is no group of transformations over jaguars (or their genes) that leaves anything invariant, but the presence of such a group is the key fact that leads to the use of the term "symmetry" in subatomic physics.

I can see the implicit analogy you're making here, but I'm not sure it's a very fruitful one. And there is one obvious disanalogy: in neutral drift, the "potential" (fitness) of different alleles of a gene is the same, whereas in spontaneous symmetry breaking, the potential of the final state is lower than that of the initial state.

One could just as well ask why we don't use the word "field" to describe jaguars or hyenas, since we use it in subatomic physics. Different areas of science use different language because they're talking about different things. The fact that there might be (strained) analogies between certain aspects of those things does not mean we should use the same word to describe them all. If you want a more general word to describe the category of things of which jaguars, hyenas, and subatomic quantum fields are all special cases, well, you just gave one: "emergent entities".

Again, I think this is not a very fruitful analogy. Another obvious disanalogy is that there is no group of transformations over jaguars (or their genes) that leaves anything invariant, but the presence of such a group is the key fact that leads to the use of the term "symmetry" in subatomic physics.

Thanks for bearing with my analogy. I do see what you mean.

I had read Nowak's example as a surprising irreducible result of discrete iteration - some one thing or another always get's picked, ending equilibrium... That this could happen to two seemingly identical things does seem... odd. Seems like some difference has to be proposed. Unless one really likes to throw stuff in the bucket of "random" (I personally don't). Change implies a degree of freedom - hence some governing rule.

Your second point (if I understand it) kind of occurred to me while "walking" the spring fever afflicted dog in the woods an hour ago. It's obvious that no two jaguars are the same jaguar. But I can imagine the same really can't be said of fundamental particles (if I might carry on the analogy just a moment longer). If they have unique patterns of spot - they are invisible to us.

Do we really think that a particle that arrives at our lab today - is a member of the same set that originated in the moments after the bang? That there are n fundamental (irreducible) billiard ball particles of type x - they are never destroyed or changed they just collide and sometimes gather together to form composite entities? Or do we think such a particle is more like one unique descendant jaguar - an emergent entity (happening) of which no two are alike, having a beginning and an end, replaced by another unique but very similar entity on some fitness landscape. Or is the point - we can't tell because the degrees of freedom over which such a fitness game are played, those that would differentiate each of those particles, are inaccessible to us (so far) if they even exist. In which case I can really sort of see why you might talk about them as fundamental "fields" with "coupling constants". In some sense one might just as well propose "they" are not really "they" at all but one thing?

[Edit] To be clear in no way am I trying to infer anything religious by that last sentence/question. I was being literal - if dense.

Staff: Mentor

Do we really think that a particle that arrives at our lab today - is a member of the same set that originated in the moments after the bang? That there are n fundamental (irreducible) billiard ball particles of type x - they are never destroyed or changed they just collide and sometimes gather together to form composite entities?

This seems like you may have a misunderstanding. Nobody is claiming that an electron detected today has been in existence since the big bang. Just that it is indistinguishable from one. If I gave you two electrons, one of which was "primordial" and the other of which was "newly minted" there would be no distinguishing feature that you could use to determine which was which. Many processes create electrons and many processes destroy them, but none distinguish them.

Staff: Mentor

I had read Nowak's example as a surprising irreducible result of discrete iteration - some one thing or another always get's picked, ending equilibrium...

I don't think this is a good description of either the evolutionary case or the subatomic physics case.

In the evolutionary case, since the drift is neutral, there is no change in fitness, and therefore no change in "potential"; so the drift is not either establishing or ending any "equilibrium". ("Equilbrium" in the evolutionary sense can only mean "optimal fitness with respect to the current environment".)

In the subatomic physics case, spontaneous symmetry breaking moves the system from a state of unstable equilibrium to a state of stable equilibrium. It does not end any equilibrium.

I think, unfortunately, that you are simply trying to use the wrong conceptual tools to reason about both of these domains.

You're right; unlike jaguars, fundamental particles of the same species really are indistinguishable. What's more, this indistinguishability has observable consequences: the statistics of quantum particles is different from what it would be if the particles were distinguishable. The current explanation for this indistinguishability is that "particles" are really just particular states of quantum fields; they're not fundamental objects. So "particles" of the same species are indistinguishable because they are states of the same fields.

In which case I can really sort of see why you might talk about them as fundamental "fields" with "coupling constants". In some sense one might just as well propose "they" are not really "they" at all but one thing?

This is another case where you are simply using the wrong conceptual tools. A "quantum field" is a particular mathematical object in a particular theory; so is a "coupling constant". Trying to reason about these things using the ordinary language connotations of the words is not going to work. To really understand them, you need to understand the underlying math.

In the evolutionary case, since the drift is neutral, there is no change in fitness, and therefore no change in "potential"; so the drift is not either establishing or ending any "equilibrium". ("Equilbrium" in the evolutionary sense can only mean "optimal fitness with respect to the current environment".)

In the subatomic physics case, spontaneous symmetry breaking moves the system from a state of unstable equilibrium to a state of stable equilibrium. It does not end any equilibrium.

Does it mean anything to ask what "causes" spontaneous symmetry breaking. If it was in an equilibrium what made it unstable? And doesn't the unstable equilibrium end in the regular old sense of "end".

I can imagine in the evolutionary case we classify some kinds of genetic mutation as "neutral" but that doesn't mean that what caused them has nothing to do with any kind of fitness. It just isn't fitness of a specific kind (some entity's reproductive fitness). And isn't the cause of the ones that get classified as beneficial to that entity's fitness exactly the same as the ones that get categorized as noise. It's only in hindsight and w/respect to some specific entity that the kind of evolutionary fitness you are talking about means anything. It's a narrower definition than I took Nowak to be talking about. His book is almost all math by the way. He's just describing the characteristics of discrete repetitive/iterative games. He might call one "repeat prisoners dilemma" but then he goes straight into the math - which to me seems to generalize specifically... As only math does (to your point).

I mean Isn't there always a "fitness function" of some kind involved in any change - are we saying that spontaneous symmetry breaking is an outcome that doesn't have one?

Staff: Mentor

An unstable equilibrium is like a pencil balanced on its point: once any tiny fluctuation takes the system away from the equilibrium point, the natural dynamics of the system takes it further away.

A stable equilibrium is like a ball at the bottom of a hole or a spring at precisely its natural unstressed length: if tiny fluctuations take the system away from the equilibrium point, the natural dynamics of the system bring it back again.

I can imagine in the evolutionary case we classify some kinds of genetic mutation as "neutral" but that doesn't mean that what caused them has nothing to do with any kind of fitness.

Yes, it does; that's the definition of "neutral" mutations.

Also, this is a physics forum, not a biology forum, and this subthread about evolutionary theory is getting way off topic, particularly since I've already pointed out that the analogy you are trying to make is not a good one. It would be best to just drop it here; if you want to discuss Nowak's book, please start a new thread in the appropriate biology forum.